We conduct Monte Carlo experiments under the model of random censorship to investigate the small sample behavior of different standard deviation formulas for the Kaplan-Meier estimator. The simulation studies include the Greenwood formula, Peto's formula and an alternative modified Greenwood formula. The results of the simulation showed that the Greenwood formula underestimated the true variance more than 90% of the times. Peto's formula overestimated the true variance when the survival rates were less than or equal to 0.5. The accuracy of the three different variance estimators is investigated using the relative bias, mean square error and confidence coefficient criteria. The absolute value of the relative bias for the modified Greenwood formula was the smallest among all three different variance estimators. On the basis of the mean square error criteria, the differences were negligible between the three different variance estimators. The Greenwood formula did not approach the confidence coefficient 0.95 in any censored observation. The confidence coefficient of Peto's formula was closer to 0.95 than the other two variance estimators. The sample sizes, censorship levels and distributions of the random variable all influenced the accuracy of the variance estimators.